A Tight VC-Dimension Analysis of Clustering Coresets with Applications
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arXiv
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| Autori principali: | , , , , |
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| Natura: | Preprint |
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2025
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| _version_ | 1866910780817932288 |
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| author | Cohen-Addad, Vincent Draganov, Andrew Russo, Matteo Saulpic, David Schwiegelshohn, Chris |
| author_facet | Cohen-Addad, Vincent Draganov, Andrew Russo, Matteo Saulpic, David Schwiegelshohn, Chris |
| contents | We consider coresets for $k$-clustering problems, where the goal is to assign points to centers minimizing powers of distances. A popular example is the $k$-median objective $\sum_{p}\min_{c\in C}dist(p,C)$. Given a point set $P$, a coreset $Ω$ is a small weighted subset that approximates the cost of $P$ for all candidate solutions $C$ up to a $(1\pm\varepsilon )$ multiplicative factor. In this paper, we give a sharp VC-dimension based analysis for coreset construction. As a consequence, we obtain improved $k$-median coreset bounds for the following metrics:
Coresets of size $\tilde{O}\left(k\varepsilon^{-2}\right)$ for shortest path metrics in planar graphs, improving over the bounds $\tilde{O}\left(k\varepsilon^{-6}\right)$ by [Cohen-Addad, Saulpic, Schwiegelshohn, STOC'21] and $\tilde{O}\left(k^2\varepsilon^{-4}\right)$ by [Braverman, Jiang, Krauthgamer, Wu, SODA'21].
Coresets of size $\tilde{O}\left(kd\ell\varepsilon^{-2}\log m\right)$ for clustering $d$-dimensional polygonal curves of length at most $m$ with curves of length at most $\ell$ with respect to Frechet metrics, improving over the bounds $\tilde{O}\left(k^3d\ell\varepsilon^{-3}\log m\right)$ by [Braverman, Cohen-Addad, Jiang, Krauthgamer, Schwiegelshohn, Toftrup, and Wu, FOCS'22] and $\tilde{O}\left(k^2d\ell\varepsilon^{-2}\log m \log |P|\right)$ by [Conradi, Kolbe, Psarros, Rohde, SoCG'24]. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2501_06588 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | A Tight VC-Dimension Analysis of Clustering Coresets with Applications Cohen-Addad, Vincent Draganov, Andrew Russo, Matteo Saulpic, David Schwiegelshohn, Chris Computational Geometry Data Structures and Algorithms Machine Learning We consider coresets for $k$-clustering problems, where the goal is to assign points to centers minimizing powers of distances. A popular example is the $k$-median objective $\sum_{p}\min_{c\in C}dist(p,C)$. Given a point set $P$, a coreset $Ω$ is a small weighted subset that approximates the cost of $P$ for all candidate solutions $C$ up to a $(1\pm\varepsilon )$ multiplicative factor. In this paper, we give a sharp VC-dimension based analysis for coreset construction. As a consequence, we obtain improved $k$-median coreset bounds for the following metrics: Coresets of size $\tilde{O}\left(k\varepsilon^{-2}\right)$ for shortest path metrics in planar graphs, improving over the bounds $\tilde{O}\left(k\varepsilon^{-6}\right)$ by [Cohen-Addad, Saulpic, Schwiegelshohn, STOC'21] and $\tilde{O}\left(k^2\varepsilon^{-4}\right)$ by [Braverman, Jiang, Krauthgamer, Wu, SODA'21]. Coresets of size $\tilde{O}\left(kd\ell\varepsilon^{-2}\log m\right)$ for clustering $d$-dimensional polygonal curves of length at most $m$ with curves of length at most $\ell$ with respect to Frechet metrics, improving over the bounds $\tilde{O}\left(k^3d\ell\varepsilon^{-3}\log m\right)$ by [Braverman, Cohen-Addad, Jiang, Krauthgamer, Schwiegelshohn, Toftrup, and Wu, FOCS'22] and $\tilde{O}\left(k^2d\ell\varepsilon^{-2}\log m \log |P|\right)$ by [Conradi, Kolbe, Psarros, Rohde, SoCG'24]. |
| title | A Tight VC-Dimension Analysis of Clustering Coresets with Applications |
| topic | Computational Geometry Data Structures and Algorithms Machine Learning |
| url | https://arxiv.org/abs/2501.06588 |